No real archaeologist would keep or sell items retrieved. The ethics of the profession require archaeologists to place artifacts in a professional repository for future analysis/research. Conditions of permitting require this to occur as well.
There are those who call themselves archaeologist who dig for "treasure" and either keep or sell the best, but they are not archaeologist in any true sense of the word. The problem with what they do is that while archaeologists excavate sites carefully to retrieve the information they holds, reading the deposits as one would read a book, bottle diggers/pot hunters cast aside everything except what they regard as collectables as though they are tearing the best plates from a rare book, only to cast aside the unread pages. In the case of public lands or private property - when operating without permission - this is trespass and/or theft.
Contrary to the exploits of famous archaeologists in media we do not personally keep, or sell, what we uncover on archaeological digs. People who choose to sell our collective cultural heritage for profit are referred to as looters, and are generally reviled in our discipline because they actively destroy our ability to understand the past.
Don't loot, please. Not even a little.
So if you aren't Indiana Jones, what is it like to be an academic archaeologist?
At the beginning everything starts out very boring. You scour through the library, and field notes from previous digs, to find the most likely location of your specific "something" interesting. Sometimes you stand on the shoulders of your advisors' sites and know "something" cool will most likely be found at location XYZ. Sometimes new building construction unearths "something" cool, but usually that type of salvage archaeology falls under the purview of Cultural Resource Management archaeologists. Sometimes you go out alone on an initial survey of the area to see if you can find any evidence of that specific "something" you are looking for.
Second, you write grants to funding agencies begging for money to support your dig. You are usually rejected, so you re-write those grants, and re-write again until a funding agency provides sufficient funds to cover the cost of conducting an excavation. All this re-writing typically takes place during the school year while you are T.A.-ing or teaching several courses.
If the grant is accepted, you hire trained research assistants to help teach excavation skills to the impressionable undergrads you've convinced into spending their summer digging in the dirt with minimal creature comforts. You typically dig in the summer, trying madly to make the best use of the precious funds and time, before returning home. As the Primary Investigator your actual fun time in the dirt is limited because you need to oversee the logistics of keeping the camp running smoothly (acquiring permits, securing ongoing access, buying supplies, etc.), but you may be called over for a particularly tricky point of the excavation, or to take pictures with the press if something awesome is found.
On site everyone has a job, and you usually trade off so each person has some measure of experience. Some people excavate a unit, some people photograph/draw unit excavations, some draw and measure finds, some map the site, some enter the data into the computer, some clean the finds and prepare them for long-term storage, etc. You take meticulous notes. These will be your life line to making sense of the season once you return home. If the PI didn't hire a cook, and you are far from town, someone will cook dinner, while others clean dishes. At the end of the day, after the chores and artifact processing are finished, you can relax at basecamp. Usually there is beer. Lots of beer.
When you find "something" cool the next steps vary. Most finds in another country stay in that country to be integrated into the local museum collections so future archaeologists can learn from the fruits of your labor. On occasion, possession and access to important finds can cause no small measure of professional rivalries and bitterness. In physical anthropology we are trying to become more open access, but it is a work in progress.
After the field season is over you pour over your notes, trying to reconcile what you found with what previous archaeologists published about the culture of interest. You write (and write, and write) during the school year in addition to teaching classes and other professional duties. You try to publish as quickly as possible so others can learn from your finds, and you can secure future funding based on previous success. Your initial drafts are rejected so you re-write again until your article is accepted. You may be writing about this field season for decades as you build upon your understanding of the site so those field notes are vitally important.
So, what does academic archaeology look like? Years of research and writing devoted to those few precious days each summer when you can play in the dirt.
No. However, if found on land owned by themselves or where they are given permission by the owner to own certain found objects (very limited conditions), they can keep or sell what they unearth. If you dig in your own backyard, where I live, you can keep or sell what you unearth.
Nearly every nation on Earth has now signed agreements strictly regulated artifacts of a particular antiquity (although, again, if on private land, virtually no enforcement).
Professional archaeologists do not typically keep or sell their finds because it ruins their scientific value. In many places, construction sites employ archaeologists to decide whether things dug up are in fact archaeological finds (if yes, they go to the research institute who is, by law, in charge of that particular area).
All parts of the U.S. and Canada are regulated by laws regarding archaeology (and while the actual definition of an artifact can sometimes be controversial, both places say that if it is pre-Columbian, it must go to the research institute that has the responsibility for that particular area).
